Integrand size = 16, antiderivative size = 45 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}} \]
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Time = 0.01 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}}+\frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {2 \int \frac {1}{\sqrt {x} (a-b x)^{3/2}} \, dx}{3 a} \\ & = \frac {2 \sqrt {x}}{3 a (a-b x)^{3/2}}+\frac {4 \sqrt {x}}{3 a^2 \sqrt {a-b x}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {2 \sqrt {x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.56
method | result | size |
gosper | \(\frac {2 \sqrt {x}\, \left (-2 b x +3 a \right )}{3 \left (-b x +a \right )^{\frac {3}{2}} a^{2}}\) | \(25\) |
default | \(\frac {2 \sqrt {x}}{3 a \left (-b x +a \right )^{\frac {3}{2}}}+\frac {4 \sqrt {x}}{3 a^{2} \sqrt {-b x +a}}\) | \(34\) |
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none
Time = 0.23 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=-\frac {2 \, {\left (2 \, b x - 3 \, a\right )} \sqrt {-b x + a} \sqrt {x}}{3 \, {\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} \]
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Result contains complex when optimal does not.
Time = 1.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 4.38 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\begin {cases} - \frac {6 a}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} + \frac {4 b x}{- 3 a^{3} \sqrt {b} \sqrt {\frac {a}{b x} - 1} + 3 a^{2} b^{\frac {3}{2}} x \sqrt {\frac {a}{b x} - 1}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\\frac {6 i a b}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} - \frac {4 i b^{2} x}{- 3 a^{3} b^{\frac {3}{2}} \sqrt {- \frac {a}{b x} + 1} + 3 a^{2} b^{\frac {5}{2}} x \sqrt {- \frac {a}{b x} + 1}} & \text {otherwise} \end {cases} \]
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none
Time = 0.19 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.67 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {2 \, {\left (b - \frac {3 \, {\left (b x - a\right )}}{x}\right )} x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a^{2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (33) = 66\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.13 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {8 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} \sqrt {-b} b^{2}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} {\left | b \right |}} \]
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Time = 0.42 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.24 \[ \int \frac {1}{\sqrt {x} (a-b x)^{5/2}} \, dx=\frac {6\,a\,\sqrt {x}\,\sqrt {a-b\,x}-4\,b\,x^{3/2}\,\sqrt {a-b\,x}}{3\,a^4-6\,a^3\,b\,x+3\,a^2\,b^2\,x^2} \]
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